A028356 -id:A028356 - cp's OEIS frontend

A028355 How the astronomical clock ("Orloj") in Prague would strike 1,2,3,...,24,25,.. (digits follow 12343212343... (A028356), n-th group adds to n).

1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212, 34321, 23432, 123432, 1234321, 2343212, 3432123, 4321234, 32123432, 123432123, 43212343, 2123432123, 432123432, 1234321234, 32123432123, 43212343212

Offset: 1

N. J. A. Sloane, J. H. Conway

This remarkable sequence is really a sequence of lists rather than numbers.

			1, 2, 3, 4, 3+2=5, 1+2+3=6, 4+3=7, 2+1+2+3=8, 4+3+2=9, 1+2+3+4=10, 3+2+1+2+3=11, 4+3+2+1+2=12, 3+4+3+2+1=13, 2+3+4+3+2=14, 1+2+3+4+3+2=15, ...

Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.

Seiichi Manyama, Table of n, a(n) for n = 1..2500
Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A028354, A028356, A068962, A118382, A118383.

s[i_] := {1, 2, 3, 4, 3, 2}[[Mod[i, 6, 1]]];
m[k_] := If[k == 1, 0, For[m0 = 1, True, m0++, If[k (k - 1)/2 == Sum[s[i], {i, 1, m0}], Return[m0]]]];
n[k_] := For[n0 = m[k] + 1, True, n0++, If[Sum[s[i], {i, m[k] + 1, n0}] == k, Return[n0]]];
a[k_] := a[k] = Table[s[i], {i, m[k] + 1, n[k]}] // FromDigits; Array[a, 27] (* Jean-François Alcover, Mar 14 2016 *)

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = 1000001*a(n-15) - 1000000*a(n-30) for n > 30.
G.f.: x*(100000*x^28 + 200000*x^27 + 300000*x^26 + 400000*x^25 + 320000*x^24 + 123000*x^23 + 430000*x^22 + 212300*x^21 + 432000*x^20 + 123400*x^19 + 321230*x^18 + 432120*x^17 + 343210*x^16 + 234320*x^15 + 123432*x^14 + 23432*x^13 + 34321*x^12 + 43212*x^11 + 32123*x^10 + 1234*x^9 + 432*x^8 + 2123*x^7 + 43*x^6 + 123*x^5 + 32*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(1000000*x^30 - 1000001*x^15 + 1). (End)

A028354 How the astronomical clock ("Orloj") in Prague strikes the hours (digits follow 12343212343... (A028356), n-th group adds to n).

Original entry on oeis.org

1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212, 34321, 23432, 123432, 1234321, 2343212, 3432123, 4321234, 32123432, 123432123, 43212343, 2123432123, 432123432, 1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212

Offset: 1

J. H. Conway

There is a single bell, which to indicate 5 o'clock, say, strikes thrice then twice.

Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.

Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A028355, A028356, A068962, A118382, A118383.

s[i_] := {1, 2, 3, 4, 3, 2}[[Mod[i, 6, 1]]]; m[k_] := If[ k == 1, 0, For[m0 = 1, True, m0++, If[k(k-1)/2 == Sum[ s[i], {i, 1, m0}], Return[m0]]]]; n[k_] := For[n0 = m[k]+1, True, n0++, If[Sum[s[i], {i, m[k]+1, n0}] == k, Return[n0]]]; a[k_] := a[k] = If[k>24, a[k-24], Table[ s[i], {i, m[k]+1, n[k]}] // FromDigits]; Array[a, 36] (* Jean-François Alcover, Mar 13 2016 *)

A068962 Number of successive terms of A028356 that add to n; or length of n-th term of A028355.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 3, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 9, 8, 10, 9, 10, 11, 11, 11, 11, 12, 13, 13, 13, 13, 14, 15, 14, 16, 15, 16, 17, 17, 17, 17, 18, 19, 19, 19, 19, 20, 21, 20, 22, 21, 22, 23, 23, 23, 23, 24, 25, 25, 25, 25, 26, 27, 26, 28, 27, 28, 29, 29, 29, 29, 30, 31, 31

Offset: 1

N. J. A. Sloane, Apr 08 2002

Suggested by R. K. Guy, Mar 01 2002.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A028355, A028356.

Each group of 15 terms is 6 more than the previous group.
Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-15) - a(n-16) for n > 16.
G.f.: x*(x^14 + x^10 + x^9 - x^8 + 2*x^7 - x^6 + x^5 + x^4 + 1)/(x^16 - x^15 - x + 1). (End)

More terms from Jim McCann (jmccann(AT)umich.edu), Jul 16 2002

A028357 Partial sums of A028356.

Original entry on oeis.org

1, 3, 6, 10, 13, 15, 16, 18, 21, 25, 28, 30, 31, 33, 36, 40, 43, 45, 46, 48, 51, 55, 58, 60, 61, 63, 66, 70, 73, 75, 76, 78, 81, 85, 88, 90, 91, 93, 96, 100, 103, 105, 106, 108, 111, 115, 118, 120, 121, 123, 126

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Cf. A047401. - Jeremy Gardiner, Jul 20 2013.

G.f.: ( 1+x+2*x^3+x^2 ) / ( (1+x)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Dec 15 2015

A158289 Period 18 zigzag sequence: repeat [0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1].

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5

Offset: 0

Jaroslav Krizek, Mar 15 2009

A toothed or zigzag sequence.
Sequence contains only numbers 0..9; abs(a(n+1)-a(n)) = 1.
Decimal expansion of 12345679/1000000001. - Elmo R. Oliveira, Feb 20 2024

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,-1,1).

Cf. A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2).

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), this sequence (k=18).

[ s lt 9 select r else 9-r where r is n mod 9 where s is n mod 18: n in [0..104] ]; // Klaus Brockhaus, Sep 07 2009

S:=[]; a:=0; for n in [0..104] do Append(~S, a); if n mod 18 eq 0 then d:=1; else if n mod 9 eq 0 then d:=-1; end if; end if; a+:=d; end for; S; // Klaus Brockhaus, Sep 07 2009

&cat[[0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]: n in [0..5]]; // Vincenzo Librandi, Jul 26 2015

a[n_] := If[m = Mod[n, 18]; m <= 9, m, 18-m]; Table[a[n], {n, 0, 85}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{}, 100, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1}] (* Vincenzo Librandi, Jul 26 2015 *)

a(n)=abs(n-round(n/18)*18) \\ M. F. Hasler, Jul 27 2015

a(18*k+j) = a(18*(k+1)-j) = j for k >= 0, j = 0..9.
G.f.: x*(1+x+x^2)*(1+x^3+x^6)/((1-x)*(1+x)*(1-x+x^2)*(1-x^3+x^6)). - Klaus Brockhaus, Sep 07 2009
a(n) = Sum_{i=0..n-1} (-1)^floor(i/9). - Wesley Ivan Hurt, Jul 25 2015
a(n) = abs(n - 18*round(n/18)). - Wesley Ivan Hurt, Dec 10 2016
a(n) = a(n-18) for n >= 18. - Wesley Ivan Hurt, Sep 07 2022

Edited and extended by Klaus Brockhaus, Sep 07 2009

A068073 Period 4 sequence [ 1, 2, 3, 2, ...].

Original entry on oeis.org

1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1

Offset: 0

Robert G. Wilson v, Mar 01 2002

Continued fraction expansion of (2+sqrt(14))/4. - Klaus Brockhaus, May 01 2010

The sequence is like a sawtooth wave of period 4. - Michael Somos, Feb 13 2011

			G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 2*x^7 + x^8 + 2*x^9 + ...

Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000034, A000120, A001045, A028356, A164356.

Cf. A177033 (decimal expansion of (2+sqrt(14))/4). - Klaus Brockhaus, May 01 2010

CoefficientList[ Series[(1 + 2x + 3x^2 + 2x^3)/(1 - x^4), {x, 0, 85}], x]
a[ n_] := {2, 3, 2, 1}[[Mod[n, 4, 1]]]; (* Michael Somos, Apr 17 2015 *)
PadRight[{},120,{1,2,3,2}] (* Harvey P. Dale, Jun 13 2020 *)

{a(n) = [1, 2, 3, 2] [n%4 + 1]}; /* Michael Somos, Feb 13 2011 */

{a(n) = n%4 + 1 - 2 * (n%4 == 3)}; /* Michael Somos, Feb 13 2011 */

{a(n) = 2 + kronecker( -4, n-1)}; /* Michael Somos, Feb 13 2011 */

G.f.: (1 + 2*x + 3*x^2 + 2*x^3) / (1 - x^4).
Conjecture: a(n) = Sum_{k=0..n} e^(i*Pi*(A000120(A001045(n)) - A001045(A000120(n)))), i=sqrt(-1). - Paul Barry, Jan 14 2005
From Paul Barry, Jan 14 2005: (Start)
G.f.: (1 + x + 2x^2)/(1 - x + x^2 - x^3);
a(n) = 2 - cos(Pi*n/2). (End)
Moebius transform is length 4 sequence [2, 1, 0, -2]. - Michael Somos, Feb 13 2011
a(n) = 2 - A056594(n). - Bruno Berselli, Mar 10 2011
a(n) = a(-n) = a(n+4) for all n in Z. - Michael Somos, Apr 17 2015
2 * a(n) = A164356(n) unless n=0. - Michael Somos, Apr 17 2015
G.f.: 1 / (1 - 2*x / (1 + x / (2 - 5*x / (1 + 16*x / (5 - x))))). - Michael Somos, Jan 20 2017
G.f.: 2 / (1 - x) - 1 / (1 + x^2). - Michael Somos, Jan 07 2019
a(n) = abs(((n+2) mod 4)-2) + 1. - Daniel Jiménez, Jan 14 2023

A175922 Period 5: repeat [1, 1, 2, -1, 2].

Original entry on oeis.org

1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2

Offset: 1

Jaroslav Krizek, Oct 17 2010

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A040001, A028356, A142150, A000012, A175921.

Flat(List([1..30],n->[1,1,2,-1,2])); # Muniru A Asiru, Sep 28 2018

&cat[[1, 1, 2, -1, 2]^^20]; // Vincenzo Librandi, Sep 28 2018

seq(op([1,1,2,-1,2]),n=1..30); # Muniru A Asiru, Sep 28 2018

PadRight[{}, 100, {1, 1, 2, -1, 2}] (* Vincenzo Librandi, Sep 28 2018 *)

a(n)=[2,1,1,2,-1][n%5+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = 1 + (2/5)*(cos(2*n*Pi/5) + cos(4*n*Pi/5) - 2*cos(2*(n+1)*Pi/5) - sin((4*n+3)*Pi/10) + 2*sin((8*n+3)*Pi/10) + sin((8*n+1)*Pi/10)). - Wesley Ivan Hurt, Sep 27 2018
G.f.: x*(1 + x + 2*x^2 - x^3 + 2*x^4) / (1 - x^5). - Vincenzo Librandi, Sep 28 2018
a(n) = a(n-5). - Wesley Ivan Hurt, Jun 25 2022

Edited by Joerg Arndt, Sep 16 2013

A177924 Decimal expansion of (28+sqrt(2730))/56.

Original entry on oeis.org

1, 4, 3, 3, 0, 2, 5, 0, 3, 4, 1, 1, 5, 2, 2, 3, 6, 6, 5, 9, 6, 0, 6, 1, 1, 9, 5, 3, 4, 6, 6, 7, 8, 7, 4, 7, 3, 1, 1, 5, 6, 4, 8, 9, 9, 1, 4, 1, 0, 3, 8, 9, 4, 9, 4, 8, 8, 5, 4, 9, 3, 5, 1, 7, 6, 7, 4, 0, 2, 9, 5, 0, 7, 6, 3, 9, 3, 6, 2, 7, 6, 0, 8, 7, 3, 5, 1, 8, 6, 9, 4, 3, 6, 9, 8, 2, 6, 4, 4, 1, 4, 6, 5, 8, 0

Offset: 1

Klaus Brockhaus, May 15 2010

Continued fraction expansion of (28+sqrt(2730))/56 is A028356.

			(28+sqrt(2730))/56 = 1.43302503411522366596...

Cf. A177925 (decimal expansion of sqrt(2730)), A028356 (repeat 1, 2, 3, 4, 3, 2).

RealDigits[(28+Sqrt[2730])/56,10,120][[1]] (* Harvey P. Dale, Mar 19 2023 *)

A187601 n/2 times period 6 sequence [1, 2, 3, 4, 3, 2, ...].

Original entry on oeis.org

0, 1, 3, 6, 6, 5, 3, 7, 12, 18, 15, 11, 6, 13, 21, 30, 24, 17, 9, 19, 30, 42, 33, 23, 12, 25, 39, 54, 42, 29, 15, 31, 48, 66, 51, 35, 18, 37, 57, 78, 60, 41, 21, 43, 66, 90, 69, 47, 24, 49, 75, 102, 78, 53, 27, 55, 84, 114, 87, 59, 30, 61, 93, 126, 96, 65, 33, 67, 102

Offset: 0

Bruno Berselli, Mar 11 2011

A007310 is a subsequence.

Bruno Berselli, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-2,4,-2,-1,2,-1).

Cf. A186813.

Cf. A109044, A088439 (by Superseeker).

[(n/2)*[1, 2, 3, 4, 3, 2][n mod 6 + 1]: n in [0..68]]; /* Other: */ [n*(5-2*(-1)^Floor((n+1)/3)-(-1)^n)/4: n in [0..68]];

CoefficientList[Series[x (1 + x + x^2 - x^3 + x^4 + x^5 + x^6) / ((1 - x)^2 (1 + x)^2 (1 - x + x^2)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,-1,-2,4,-2,-1,2,-1},{0,1,3,6,6,5,3,7},90] (* Harvey P. Dale, Aug 20 2017 *)

a(n) = (n/2)*A028356(n).
G.f.: x*(1+x+x^2-x^3+x^4+x^5+x^6)/((1-x)^2*(1+x)^2*(1-x+x^2)^2).
a(-n) = -a(n).  a(n) =  2*a(n-1)-a(n-2)-2*a(n-3)+4*a(n-4)-2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8) for n>7.
a(n) = n*(5-2*(-1)^floor((n+1)/3)-(-1)^n)/4.

A164360 Period 3: repeat [5, 4, 3].

Original entry on oeis.org

5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3

Offset: 0

Stephen Crowley, Aug 14 2009

From Klaus Brockhaus, May 29 2010: (Start)
Continued fraction expansion of (32+sqrt(1297))/13.
Decimal expansion of 181/333. (End)

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A007877 (repeat 0,1,2,1), A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2), A158289 (repeat 0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1).

Cf. A178566 (decimal expansion of (32+sqrt(1297))/13). [Klaus Brockhaus, May 29 2010]

[ n le 3 select 6-n else Self(n-3):n in [1..105] ]; // Klaus Brockhaus, Sep 17 2009

&cat [[5, 4, 3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([5, 4, 3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{}, 100, {5, 4, 3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n) = 4+(-1)^n*((1/2+I*sqrt(3)/6)*((1+I*sqrt(3))/2)^n+(1/2-I*sqrt(3)/6)*((1-I*sqrt(3))/2)^n). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = 4+(1/3)*sqrt(3)*sin(2*n*Pi/3)+cos(2*n*Pi/3). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = a(n-3) for n > 2, with a(0) = 5, a(1) = 4, a(2) = 3.
G.f.: (5+4*x+3*x^2)/((1-x)*(1+x+x^2)). [Klaus Brockhaus, Sep 17 2009]
E.g.f.: 4*exp(x)+(1/3)*sqrt(3)*exp(-(1/2)*x)*sin((1/2)*x*sqrt(3))+exp(-(1/2)*x)*cos((1/2)*x*sqrt(3)).
a(n) = 4 + A057078(n). - Wesley Ivan Hurt, Jul 01 2016

Edited by Klaus Brockhaus, Sep 17 2009

Offset changed to 0 and formulas adjusted by Klaus Brockhaus, May 18 2010

cp's OEIS Frontend

A028355 How the astronomical clock ("Orloj") in Prague would strike 1,2,3,...,24,25,.. (digits follow 12343212343... (A028356), n-th group adds to n).

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A028354 How the astronomical clock ("Orloj") in Prague strikes the hours (digits follow 12343212343... (A028356), n-th group adds to n).

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A068962 Number of successive terms of A028356 that add to n; or length of n-th term of A028355.

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A028357 Partial sums of A028356.

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A158289 Period 18 zigzag sequence: repeat [0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1].

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A068073 Period 4 sequence [ 1, 2, 3, 2, ...].

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A175922 Period 5: repeat [1, 1, 2, -1, 2].

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